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Realising wreath products of cyclic groups as Galois groups. (English) Zbl 0662.12010
Let K be a field of characteristic zero, T, X indeterminates algebraically independent over K. For \(n\geq 1\), let k(n)\(\geq 2\) be an integer, \(f_ n(X,T)=X^{k(n)}+T\). Put \(F_ 1(X,T)=f_ 1(X,T)\) and define for \(n\geq 1\), \(F_{n+1}(X,T)=F_ n(f_{n+1}(X,T),T).\)
In theorem 1 the author proves that if \(\overline{K}\) is the algebraic closure of K then the Galois group of \(F_ n(X,T)\) over \(\overline{K}(T)\) is isomorphic to the wreath product \(\Gamma_ n=G_ 1[...[G_ n]...]\) where for each \(i\leq n\), \(G_ i\) is the cyclic group of order k(i) with its natural permutation action on the symbols 1,...,k(i).
As a corollary the author proves that if K is a Hilbertian field containing the k(i)-th roots of 1 for \(i\leq n\) then given \(t>1\) there is a finite Galois extension L over K such that Gal(L/K) is isomorphic to the direct product of t copies of \(\Gamma_ n\).
Reviewer: T.Soundararajan

MSC:
11R32 Galois theory
12F10 Separable extensions, Galois theory
20F29 Representations of groups as automorphism groups of algebraic systems
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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References:
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[6] DOI: 10.2996/kmj/1138036122 · Zbl 0435.12010 · doi:10.2996/kmj/1138036122
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[8] DOI: 10.1112/plms/s3-51.3.385 · Zbl 0622.12011 · doi:10.1112/plms/s3-51.3.385
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